In mathematics, Abel's theorem for power series relates a limit of a power series to the sum of its coefficients. It is named after Norwegian mathematician Niels Henrik Abel.
Contents
Theorem
Let
be a power series with real coefficients ak with radius of convergence 1. Suppose that the series
The same theorem holds for complex power series
for some M. Without this restriction, the limit may fail to exist: for example, the power series
converges to 0 at z = 1, but is unbounded near any point of the form eπi/3n, so the value at z = 1 is not the limit as z tends to 1 in the whole open disk.
Note that G(z) is continuous on the real closed interval [0, t] for t < 1, by virtue of the uniform convergence of the series on compact subsets of the disk of convergence. Abel's theorem allows us to say more, namely that G(z) is continuous on [0, 1].
Remarks
As an immediate consequence of this theorem, if z is any nonzero complex number for which the series
in which the limit is taken from below.
The theorem can also be generalized to account for infinite sums. If
then the limit from below
Applications
The utility of Abel's theorem is that it allows us to find the limit of a power series as its argument (i.e.
Outline of proof
After subtracting a constant from
Given
when z lies within the given Stolz angle. Whenever z is sufficiently close to 1 we have
so that
Related concepts
Converses to a theorem like Abel's are called Tauberian theorems: There is no exact converse, but results conditional on some hypothesis. The field of divergent series, and their summation methods, contains many theorems of abelian type and of tauberian type.